1. Field of the Invention
The invention relates to a method to detect and/or determine the regularity of a cyclically fluctuating scientific or technical quantity as well as to a device to perform the method.
2. Description of the Related Art
A fluctuating quantity is a quantity which oscillates as function of time. A cycle is the time span between three successive crossings of the average of a scientific or technical quality (technical QUANTITY-A). A crossing of the average is a point in time when the technical QUANTITY-A equals the average of technical QUANTITY-A.
The regularity of behavior of technical QUANTITY-A may include an impact from other cyclically fluctuating scientific or technical quantities. Examples of cyclically fluctuating scientific or technical quantities are the amount of fuel injected into a combustion engine, the resulting torque or respiratory activity and heartbeat length of humans.
The detection and determination of behavioral regularity is related to phenomenological reconstruction methods of multivariate processes. This is often achieved by using linear approaches (xe2x80x9cBox G. E. and G. M. Jenkins, Time series analysis: forecasting and control, Holden-Day, Oakland, Calif., 1976xe2x80x9d). In addition to this, nonlinear approaches have been chosen which approximate the behavioral regularity, e.g., by polynomials or by so called radial basis functions (xe2x80x9cCasdagli M., Nonlinear prediction of chaotic time series, Physica D 35, 335, 1989xe2x80x9d and xe2x80x9cBroomhead D. and Lowe D., Multivariable functional interpolation and adaptive networks, Complex Syst. 2, 321, 1988xe2x80x9d).
In the case of a regularity involving several quantities, synchronous or temporarily synchronous behavior is of special interest. The notion of synchronization, which was originally limited to periodically oscillating quantities, has recently been generalized to aperiodically fluctuating quantities. Phase synchronization (Rosenblum M. G., A. S. Pikovsky und J. Kurths, Phase Synchronization of chaotic oscillators, Phys. Rev. Lett. 76, 1804, 1996) is a bivariate regularity, which can even be determined in the case of fluctuating quantities with instationary amplitudes.
The analysis of phase synchronization is often based on the so-called Hilbert phase (Rosenblum et al., 1996). In general a regularity expressed in terms of Hilbert phases cannot be transformed to a regularity expressed in terms of the fluctuating original quantities. One reason for this is the fact that, without special precautions, a reconstructed trajectory in the phase or angle space may pass the immediate neighborhood of a singularity of the functional determinant of the back-transformation to the original (Cartesian) coordinates.
When developing tools to analyze coupled, aperiodic and instationary quantities, cardio-respiratory interaction plays the role of a pilot study. As expressed in xe2x80x9cHirsch J. A, and B. Bishop, Respiratory sinus arrhythmia in humans: how breathing pattern modulates heart rate, Am. J. Physiol. 241: H620-H629, 1981xe2x80x9d, respiratory activity and heartbeat length fluctuate in xe2x80x9csynchronyxe2x80x9d. In addition xe2x80x9cPessenhofer H. und T. Kenner, Zur Methodik der kontinuierlichen Bestimmung der Phasenbeziehung zwischen Herzschlag und Atmung, Plxc3xcgers Arch. 355, 77:83, 1975xe2x80x9d describe a so called phase locking phenomena which result from the fact that heartbeats have the tendency to occur preferentially during certain phases of the respiratory cycle.
Both forms of cardio-respiratory interaction are seen as an indicator of a healthy enervation of the heart. Absence of synchronization in a state of physical and psychological relaxation may be interpreted either as a symptom of an autonomous neuropathy (e.g., Diabetes mellitus) or as an increased risk of heart attack.
German patent application 197 18 806.0 to describes a device and method to determine the regularity of the heartbeat rate. The device shows a pacemaker to trigger the respiratory activity as well as a detector to measure the heartbeat rate. The device determines a behavioral regularity of the heartbeat lengths of a healthy human in relaxed state. A measure of the regularity is a two-dimensional pattern, which is determined as follows.
During voluntary paced respiration, the ECG of a subject is recorded and the heartbeat lengths are extracted. A high pass filter is applied, which eliminates frequencies lower than the breathing frequency. The sequence of heartbeat lengths is converted to a two-dimensional graph showing the length of a heartbeat as function of the preceding heartbeat length. A typical graph will result in an ellipse type pattern.
The coordinates of this graph are transformed to polar coordinates (angles and radii) using the average heartbeat length as the center. This way, a certain number of heartbeat lengths are converted to a set of angles, which may be interpreted as an alternative to the Hilbert phase (and whose number is reduced by one).
The sequence of angles is also converted to a two-dimensional graph representing the these angles as function of their preceding angle.
For young subjects and slow pacing in particular, this graph results in a pattern similar to a two step staircase. In these cases, a deviation from the typical pattern may be taken as a symptom for physical or autonomic abnormality or change. Further studies have given evidence that this method is less useful for older humans. In particular, this method relies on voluntary pacing, which takes some time to perform in a relaxed state. Thus the method cannot be applied without difficulties.
The three publications, U.S. Pat. No. 5,159,249, European Patent EP 0 338 705 A and TCHON K. et al. xe2x80x9cA normal form solution to the singular inverse kinematic problem for . . . , proceedings of the IEEE International Conference on Robotics and Automation, U.S., New York, N.Y.: IEEE, Bd CONF. 15. 1998, pages 3222-3227xe2x80x9d offer three solutions to avoid singularities of the transformation from Cartesian coordinates to angle coordinates. However, the three solutions are not suited to set up a general time series model, which guarantees the avoidance of these singularities by construction. In particular, they do not provide any clues for finding appropriate alternative angle- or phase definitions.
The aim of the invention is the creation of a universal and simple method to detect and determine the regularity of the phase modulation of a cyclically fluctuating quantity including the possibility to determine the regularity induced by cross impact from other cyclically fluctuating quantities as well as the guarantee that the back-transformation of reconstructed phase modulation does not lead to singular reconstructions of the cyclically fluctuating quantity. A further aim of the invention is the deployment of a device to perform the method.
The aim is achieved by the method and device described below.